Can you bake keyframes in BGE and have them work in cycles?
In short, yes you can. It is as simple as activating a checkbox called Record Animation in the Menu, and then playing your game:



When you hit Esc to exit the game, it will take a little while, as then all keyframes are evaluated. Once done, you can select the object and edit keyframes in the Graph Editor, although it might not be great fun to do so:



Using clean channels or clean keyframes you can simplify the curve, but the risk is to change the animation too much: